reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: ZFMISC_1:81
  bool A (\/) bool B c= bool (A (\/) B)
proof
  let i be object;
  assume
A1: i in I;
  then
A2: (bool (A (\/) B)).i = bool (A.i \/ B.i) by Lm2;
  (bool A (\/) bool B).i = (bool A).i \/ (bool B).i by A1,PBOOLE:def 4
    .= bool (A.i) \/ (bool B).i by A1,Def1
    .= bool (A.i) \/ bool (B.i) by A1,Def1;
  hence thesis by A2,ZFMISC_1:69;
end;
